Question

Find the next term in the sequence \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\).

Short answer

The next term in the sequence is \(\frac{7}{2}\).

Step-by-step solution

Observe the Pattern

Examine the differences between consecutive terms in the sequence. The first term is \(\frac{1}{2}\), the second term is \(\frac{3}{2}\), and the third term is \(\frac{5}{2}\).

Calculate the Difference

Calculate the difference between the second term and the first term: \(\frac{3}{2} - \frac{1}{2}\ = 1\). Similarly, calculate the difference between the third term and the second term: \(\frac{5}{2} - \frac{3}{2} = 1\).

Identify the Rule

The differences between the terms are equal, indicating this is an arithmetic sequence with a common difference of \(1\).

Apply the Rule to Find the Next Term

Add the common difference \(1\) to the last term \(\frac{5}{2}\) to find the next term: \(\frac{5}{2} + 1 = \frac{7}{2}\).

Key concepts

Common Difference

In an arithmetic sequence, the term "common difference" is key to finding the pattern that the sequence follows. It represents the constant amount that each term increases or decreases from the previous term. This value is always the same for all consecutive terms within the sequence.
To find the common difference, simply subtract any term from the term that follows it. For instance, looking at the sequence from our exercise:

• The difference between the second term, \(\frac{3}{2}\), and the first term, \(\frac{1}{2}\), is \(1\).
• Likewise, the difference between the third term, \(\frac{5}{2}\), and the second term, \(\frac{3}{2}\), is also \(1\).

This consistent difference of \(1\) confirms that the sequence is indeed arithmetic. Understanding the common difference is essential, as it helps us determine the rule for extending the sequence, making it a fundamental concept in arithmetic sequences.

Sequence Pattern

A sequence pattern is essentially the rule that governs how terms in a sequence are generated. Identifying this pattern is crucial for predicting future terms and understanding the behavior of a sequence. In an arithmetic sequence, this pattern is defined by the common difference. For every arithmetic sequence, the pattern can be summarized in a simple form: each term is the previous term plus the common difference.
For example, our sequence \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\) follows the pattern of adding \(1\) to the previous term to obtain the next.
Recognizing the sequence pattern allows you to find any term in the series without needing to list out all previous terms. This understanding is incredibly useful when working with larger numbers of terms or more complex sequences.

Term Calculation

Term calculation in arithmetic sequences involves using both the first term of the sequence and the common difference to find any term number quickly and efficiently. This provides a reliable method for calculating any desired term, even for long sequences.
The general formula for finding the nth term in an arithmetic sequence is: \[ a_n = a_1 + (n - 1) imes d \] where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
Applying this to our example sequence:

• The first term \(a_1\) is \(\frac{1}{2}\).
• The common difference \(d\) is \(1\).

Suppose you want to find the fourth term; you'd plug these values into the formula: \[ a_4 = \frac{1}{2} + (4 - 1) \times 1 = \frac{7}{2} \] This same approach works for any term, ensuring you can efficiently and accurately calculate further terms in the sequence.